Speaking of our human identity, I have already mentioned three hard facts: (1) we are born from a mother (see lecture on Greek tragedy), (2) we die (see lecture on the Otherworld), and (3) our language cannot communicate our uniqueness (see lecture on Language). Today we add a fourth unavoidable fact: we are creatures living in space and time. Our technologies change the ways in which we conceive of space and time, but cannot eliminate the fact that we live in space and time.
First, let me briefly address the question how we, as human beings, acquire our space and time notions in babyhood and childhood. This question is being studied by various disciplines: developmental psychology, neurobiology, cognitive science. The pioneers in this kind of research were two German scientists, Johannes Peter Müller (1801-1858) and Hermann von Helmholz (1821-1894); the second one is famous for having stated the physical principle of conservation of energy, of which we'll have more to say next semester in our lectures on physics. The new horizons opened by these two scientists can be briefly summarized as follows: as I mentioned in my first lecture on geometry, up to the early 1800's the properties of space and time were thought to be innate—we are born, Kant and others thought, following the Greek philosophers of Plato's school, with a body and a mind built in such a way that the 5 Euclidean axioms hold true. As we saw in the lecture on geometry, from about 1820 to 1860, with the development of non-Euclidean geometries and the recognition of the possibility that our universe may not be Euclidean, Kant's view was discarded, and the way was opened for the investigation of the physiological processes in our eyes, limbs and ears, as well as in our nervous system, which underlie the construction of our notions of space and time. The modern view is that these notions are developed in ways similar to our acquisition of language: certain structures are innate, part of our genetic endowment; they are put to work during early life so that our brain finally becomes wired the way it is. A child never exposed to language will likely never speak, and a person born blind and entirely paralyzed will likely never have normal notions of space.
Developmental psychologists make observations of the behavior of children at different stages of life; neurobiologists experiment with animals—the horseshoe crab, for example, is a favorite because of its relatively simple neural system within a relatively large body: a video camera is attached to a male, the animal goes searching for a female in the surf, and the visual images and nerve impulses are fed into a computer to try to break the crab's neural code for vision. One thing we can say right away: the space-time notions of different species are quite different. When it comes to human beings, one of the main sources of data are people with brain disease or brain injury: in those cases one can determine, by relating the deficiency with the injury, which parts of the brain process our normal space and time relations, as well as language tasks and other functions. Roughly speaking, it seems that the left hemisphere of our brain has mostly to do with language, symbols and concept formation, while the right hemisphere has more to do with visual-spatial thinking. When we observe that young people have become much better at visual images than verbal tasks (mainly because of TV), what we might be seeing is that the right part of their brain is being stimulated and developed to the detriment of the left side. As for time notions, the right hemisphere seems to process data from the past and the present, while the left one projects toward the future, just as the two brothers of Greek legend, Epimetheus and Prometheus, whom Prof. Isser mentioned a while ago. This division of the brain in two halves occurs only in mammals, and there is an unconfirmed and rather wild theory that the two halves were not connected (as they are now) until very recently in the history of humankind, maybe 6 thousand years ago. As a matter of fact, there are people who are born and live apparently happy lives without the neural connection between the two hemispheres.
This lecture must be kept a secret between us: if it was known that I am trying to teach you about space and time in the brief space of one hour, I may be certified a lunatic. Let's go back to the modern definition of geometry, that is, to what scientists came up with after the invention of non-Euclidean geometries and the collapse of Kantian apriorism, around 1850. What is geometry? It is the study of the properties of figures (like triangles, circles, etc.) in the plane, solids (like cubes, spheres, etc.) in 3 dimensional space, and so on to higher dimensions. But (and here's a big BUT), geometry does not study ALL the properties of figures or solids—only some, some strictly specified ones. To say it at once in a technical phrase that I'll explain in a minute: Geometry is the study of those properties of figures or solids which remain invariant under the action of a group of motions; different groups of motions give rise to different geometries. Let's start with Euclidean geometry. Here's a quadrilateral, a sheet of paper as it happens, and here's another, drawn on the board. From the point of view of Euclidean geometry, these two figures are considered equal if I can move one of them and superimpose it exactly on the other. The motions I am allowed in Euclidean geometry are: translations, rotations, reflections, and any combination of those. These motions form what's called the group of Euclidean motions. The fact that this figure is made of white paper and the other is made of green board is of no concern: color, texture, material are not geometric notions. These two quadrilaterals are equal if I can move one onto the other, period. On the other hand, the length of the sides, the angles, the area of the quadrilateral, those are the notions Euclidean geometry deals with. This is what makes geometry an abstract discipline: to abstract means originally (in Latin) to draw or take away; we are taking away from this rectangle the facts that it is made of paper, that it is white, that it was made for writing on it, etc.
I mentioned groups of motions. The notion of group is one of the most important in modern science. A group of motions is a collection, a set of motions, but not just any set: two conditions must be fulfilled. First, if you follow one motion in your set by another motion in your set, the composite motion must also be in your set. Second, if you take any motion in your set, there must be another motion in your set, the inverse one, which undoes what the first did; in other words, if you perform a motion on this rectangle, say, and then follow it by the inverse motion, the rectangle comes back to where it was originally. As it turns out, there are many other groups of motions, different from the Euclidean group of translations, rotations and reflections: each group gives rise to a different geometry. In so-called projective geometry, for example, which originated with the technique of perspective in drawing, the group is larger than the Euclidean, and two rectangles which are not equal in the Euclidean sense may be equal in the projective sense. I cannot get here into a description of these other groups—suffice it to say that in this way we have covered all geometries, not merely the old Euclidean one. The larger the group, the more we abstract from reality: projective geometry is more abstract than the Euclidean, but all geometry is abstract, that is, it takes away infinitely much from our perception of reality.
To say that geometry is abstract is not to criticize it: all science is abstract. This realization marks the beginning of modern science and the rejection, in the 17th century, of the old Aristotelian scholasticism, as we will see in my next two lectures. Going back to the conception of geometry as the study of properties of objects which do not change when we move them with the motions belonging to a group, it was the great French mathematician Henri Poincaré (1854-1912) who first connected this conception to the developments in human babyhood. The experience of moving the muscles of limbs and eyes back and forth, right and left, etc., implants in our brain the mechanism of groups of motions, even though we aren't consciously aware of it. Later studies of child development of space notions, like those of the Swiss psychologist Jean Piaget, were inspired by the pioneering ideas of Poincaré.
Meanwhile, what is the role of time in all this? This can be briefly stated: time is the great abstracted, time is taken away. The word "positivism" has had various meanings in philosophy, starting with the French philosopher Auguste Comte around 1840, but I define it thus: positivism is the belief that knowledge is impossible unless we take time away from consideration. The British philosopher and logician Bertrand Russell (1872-1970), a champion of positivism, put it succinctly: "The beginning of wisdom is to realize the unimportance of time." How is time taken away? In logic, which establishes the rules for thinking, by the identity principle: "A is equal to A," everything is equal to itself, even though, as Prof. Ng mentioned in her lecture on Buddhism, we never actually experience anything that doesn't change in time. For example, the cells, the molecules and atoms in our bodies are constantly being replaced by others. In geometry, as we have seen, time is taken away or abstracted by a double disappearance act: first, we only look at properties which remain invariant, which do not change with motion; and secondly, by the definition of group, which requires every motion to be reversible, to be undone by another motion. In our experience time is not reversible, most of the time we cannot undo what we do, we cannot go back to yesterday, nor can we bring what's dead back to life.
Physics, of course, cannot do away with time. Next semester, if we study a little thermodynamics, we'll see how physics copes with the irreversibility of time. Physics makes a model of time as a geometric object, considers it as an infinite line, the so-called real number line. This means that we must fix a starting point, an origin, and start counting; the unit of measure—be it a day, an hour, a second—is always the same, never changing, and this is what allows us to assign a real number to each instant of time (remember Prof. Hagelberg's lecture on physics). We express this by saying that in physics time is homogenous, and this is what allows us to apply mathematics to physical problems. Thus time, for modern physics (that is, the physics created in the 17th century), becomes a fourth dimension: three dimensions for space and one for time. It is often said that mythical time, the time of primitive religions, is circular—all things recur after a longish period—as opposed to modern, scientific time, which is an open line; but this is not a fundamental difference: the fundamental abstraction was to make of time a geometric object, whether a circle or a line doesn't matter much. In other words, although time seems to be accepted by physics, it is subtly negated by being considered homogenous, itself not subject to change. Up to our century, the 3 dimensions of space and the one dimension of time were strictly separate—this means that the group of motions which defined the geometry used by physics moved objects in space independently from motions in time, but with Einstein's Theory of Special Relativity a new group, the Lorenz group of motions, became the norm, and now it is accepted that the length of a ruler, for example, becomes noticeably shorter when it is traveling at speeds near the speed of light. Space and time became intermingled in Relativity Theory. But, as the French philosopher Henri Bergson (1859-1941) pointed out in a polemic with Einstein, physics still made time into a spatial, geometric object, ever the same, and thus the essence of time—change and radical newness—was negated.
The negation of time is much older than the birth of modern physics. Two thousand years before that, when Plato defined time as "a movable image of eternity," he led us to understand that the real thing is timeless eternity, time being only an image, "an epiphenomenon," something we have to put up with, given our pitifully weak bodies, but which shouldn't disturb one seriously in search of wisdom. This is exactly what Bertrand Russell said in our own century, and it is the positivist's faith. As I mentioned in an earlier lecture, modern science owes much to Plato. What's at stake here is tremendous—nothing less than human freedom and the possibility of surprise. If only eternity is real and time is mere appearance, there is no freedom of the will and future time is preordained, or can be computed, given enough information about the past. Philosophers and theologians have been struggling with this problem for centuries: either they went rationally along with scientific notions of space-time and thereby had to deny freedom, or they affirmed freedom and were called irrationalists, like Bergson. A positivist should not be surprised at anything, or rather, if he calls a new scientific discovery "surprising" he would immediately add that, had he thought more deeply about it, had he been smarter, he wouldn't be surprised. Further, a true positivist believes in Progress—history is invariably proceeding toward the better, toward the "more advanced"—which is another way of denying the threatening and surprising nature of time. Incidentally, reading what you wrote in your last take-home test, especially those questions having to do with ancient texts—Genesis, Plato, Aeschylus, Gilgamesh—I noticed that some of you seem to believe that people now and societies now are superior to the ancient ones. All of you would bristle at the proposition that a race, a sex, or a nation is superior to the others, yet do not seem to be outraged at the belief that we are superior to our ancestors. Let me just say that applying concepts like superior and inferior to different cultures is uneducated. We modern people may have gained in some respects, but perhaps we have lost in other equally important respects. And one of these respects, I hope to show you, is the way we conceive of, and deal with, space and time.
But to go back to the surprising nature of time, to illustrate
how logic can deal a mortal blow to the concept of surprise, let
me tell you a paradox, as surprising as Zeno's paradoxes of which
we talked before. And, again, reading your homework, I noticed
that some of you tend to conclude that you don't believe in, or
don't agree with, what a paradox concludes. It is not a matter
of agreeing or not. Let me repeat: a paradox presents us with
a conflict between on the one hand our concepts and our logic
and on the other hand our immediate experience. A paradox is an
invitation to modify our logic and our concepts.
Suppose I tell you that next week there's going to be a surprise quiz; I'm not telling you the day—from Monday through Friday—only that it's going to be next week. Now someone says, "No Sir, you can't do that, you cannot give us a surprise quiz next week." "Why not?" "Because, if by Thursday evening you haven't given us the quiz, we will then know for sure it's going to be on Friday, so it would be no surprise; so it can't be given on Friday. Knowing that, suppose by Wednesday evening you haven't given us the quiz: since it can't be on Friday, it must be on Thursday, so again it would be no surprise—therefore it can't be given on Thursday. And so on, down to Monday. You cannot give us a surprise quiz on any day." I'll let you struggle with the logic of this, and let me warn you, it isn't easy. It all boils down to this: the concept of surprise defies logic, because logic is timeless. Another way to look at it: if you are expecting a surprise, it's not going to happen; only when the horizon remains open for both encountering and not encountering, for both being and not being, does surprise become possible.
We have talked about abstractions, mostly about the scientific or positivistic abstractions of space and time, and the consequent denial of freedom and surprise. Still, freedom and surprise are elements of our lived experience. Or at least, we dearly wish them to be, for without them life would be a joyless drag. The Romantic solution was to separate two spheres of action: Science was to deal with abstraction and precise, mathematical knowledge, while Art was to deal with, and communicate, the concrete human experience. Influenced by Bergson, Marcel Proust wrote an enormous novel, "A la recherche du temps perdu" (a very suggestive title in our context, which can be translated as: In Search of Lost Time). Proust tried to communicate his experience of non-abstract time—not wasted time, mind you, but the sense of time that's lost when we limit ourselves to scientific, positive understanding. In the usual but unusually inept English translation—Remembrance of Things Past—that meaning itself is lost. The early Romantics (end of 18th, beginning of 19th century) glorified a special function of the mind, the imagination, different from reason and the understanding. The imagination was supposed to create such things as works of art, but the very nature of the imagination, which deals with what's not there, or at least with what we don't perceive, betrays the Romantic inferiority feelings in confront with science, which deals with what's effectively there and with what we do perceive.
Of all the arts, architecture is the one which most influences our sense of space and time, because it defines our lives. It is also the most conservative of all the arts, especially the architecture of institutional buildings such as our campus. Every once in a while a book, a painting, a movie, a piece of music defies the accepted commercial canons, in spite of the enormous difficulties of getting anything original published, shown or played; but too much money is invested in institutional buildings, and too much prestige is at stake for any true innovation to show up in their design. Big American cities are dominated by modernist geometric architecture, whose main thrust is a denial of time. Official, institutional architecture has always aimed at eternity: the pyramids of Egypt are, after all, still standing, and stone, the preferred material, is far more lasting than human life. Awesome power—the power of kings, of tyrants, and also of modern institutions, governments, banks and corporations—is expressed primarily by two characters: durability and huge size. Hitler and his architect, Albert Speer, recognizing that even the most lasting structures eventually become ruins, were concerned that those ruins should be awe-inspiring. We have some huge stones, so-called "primitive" representations of the human figure or face from Eastern Island in the Pacific and from the Olmec culture in Mexico, which were probably intended to perpetuate the life of important people. These figures are quite imposing, but they do not express denial of time and defiance of death as effectively as the pyramids. What's missing is a third element, beyond durability and size, and that is symmetry.
What is symmetry? It is invariance, that is, an object is symmetric when a certain group of motions leaves it unchanged. The larger the group of motions, the more complete the symmetry. Thus, roughly speaking, the human body is bilaterally symmetric—invariant under reflection about a plane cutting us in half. But this is a very small group (only two elements); a pyramid is more symmetric, the group of motions leaving it unchanged is larger; a cube is even more symmetric, but the sphere is supremely symmetric—here the group is infinite. Symmetry being invariance, permanence, it elicits in us the satisfaction of a triumph over time and the disintegration time brings. As we have seen in our lecture on the Presocratics, Parmenides taught that the universe is a sphere eternally the same, unchanging: here time is totally negated, and with it, the idea of not-being, the idea of death.
Without relinquishing durability and hugeness, contemporary institutional architecture relies chiefly on symmetry to convey its denial of time. As I said, simple geometric objects exhibit the greatest symmetry, and this is why Philip Johnson, considered the founding father of American modern geometric architecture and the designer or inspirer of many of the tallest buildings in our cities, went back to what he thought was Plato's definition of beauty: "Absolute beauty," he wrote, "is found in straight lines and circles, and plane or solid shapes, rather than in living figures." Never mind that this is a poor, simplistic reading of Plato; the main problem with this conception of "absolute beauty" is that it is boring, there is no surprise, nothing unexpected. Just look at our campus: all straight lines, except for a few curves at the top, which, as far as I can tell, are arcs of hyperbolas, and so completely symmetric that a carillon tower had to be added off center for people to be able to orient themselves. This concept of what constitutes beauty in architecture is so widespread among U.S. practitioners that architecture tends to be thought of as a branch of applied mathematics, without any relation to human nature and human needs. Besides, the kind of mathematics these practitioners have in mind has little to do with modern developments (as you can read in the book by Ekeland): it is the kind the Pythagoreans and Plato had in mind.
The main problem with excessive symmetry, as I said, is that it is boring. But this is only a symptom of a more general fault, found in geometric architecture as well as in many other artistic works—I mean, the lack of spirituality. The word "spiritual" has been terribly abused, but in this course about human identity and technology it is my duty to define it clearly, so that it doesn't become some kind of empty hocus-pocus. Spirituality is a tension, an on-going engagement with the double nature of time, I mean with the fact that time kills us and those we love, but also opens up the possibility of surprise and the unexpected, and gives birth to the miraculous. The Buddha renounced the world when he saw sickness, old age and death, but he experienced, too, the miracle of sudden illumination under the Bodhi tree. Time, in this sense, is the very opposite of eternity: it is at once a curse and a blessing. I am not telling you how to live with this tension—there are infinitely many ways, as many as different human beings: this is not a recipe for becoming spiritual, only a criterion by which to judge what is spiritual and what is not. Now, some symmetry is necessary for the appearance of surprise. If it weren't for the fact that students come to class regularly during the week, a surprise quiz would be impossible; in works of art, as in life, something surprises us because it breaks with the ordinary symmetry of things. Thus the spiritual, the vivid consciousness of the nature of time, lies somewhere between extreme symmetry and complete chaos. This is also the region where (please read Ekeland's book carefully) the interesting mathematical facts appear, discovered by the mathematician Henri Poincaré, in what's called the dynamics of non-linear systems, popularly known as "chaos theory." In other words, the spiritual lies in a middle ground between frigid, perfect form and total chaos, and this is true also in our lives and in works of art.
Architects such as Johnson may point to Greek temples as a justification for the use of "pure forms." But again, this is a misunderstanding. First, because Greek temples are not just buildings but the combination of a building with the landscape around, which provokes a feeling of unexpected, and this is not true of the skyscrapers in our cities. Second, because Greek temples were abundantly provided with lifelike ornamentation, and most importantly, because they were painted in vivid colors, of which only traces have remained. American institutional architecture—America in general—is deadly afraid of color—color is something that changes with light and appears differently at different times of day; color, like the Buddha Prof. Ng talked about last time, laughs at the notion of permanence and eternity. So our big buildings are various shades of gray. We are color-deprived, and this is one reason why the poorest parts of our cities are more depressing than the poorest parts of the cities of countries such as Mexico or Brasil, countries much poorer than the U.S. We are being fed only with the extremes, symmetry or chaos, we are overstuffed with both symmetry and chaos, but you are not likely to notice it until you start comparing our own city spaces with those in the so-called underdeveloped countries, or with the towns which are left from older times, from the middle ages. You will not find a medieval church in Europe that's too symmetric, but go around and try to find a big church, a U.S. cathedral, which is not perfectly symmetric, to the last detail: thus those very buildings supposed to be "spiritual" are completely lacking in spirit.
What has our civilization done to space-time? We are able to move faster than ever, true. But our space and our time are inhuman, unlivable, unspiritual. One symptom is our word "inner": it used to be a highly positive spatial metaphor—people used to say "the inner heart," "the inner soul," "my innermost conviction," etc. It has become a negative word used to denote those portions of our space which put the U.S. to shame: the "inner cities." Another symptom is the following: if you look up Time in the Encyclopedia Britannica, you'll find an article on Measurement of Time, but none on time itself. If you look up Space, you'll find an article about Space Exploration and another about Space-Time in Relativity Theory, but none on space itself. Is this because no one ever said anything interesting about time and space as such? On the contrary, if we were to present the views of thinkers and artists on these topics through the ages, we could spend a lifetime doing just that; of course, we can find some of those views on the articles on those thinkers and artists in the Britannica; but again, no article on time or space as such. For the compilers of contemporary knowledge, time flows only to be measured; space is either outer space to be explored by NASA, or space-time to be explored by physics. This is a symptom of a cultural disease. Don't get me wrong: measuring time and outer space exploration are fascinating activities, and so is physics; we'll have much to say about them in my next two lectures; but even specialists, even clock-makers, astronauts and physicists, are human beings, and as such they must live in space through time—a living space, a home, a city, suburbia or countryside, a campus; and a lived time with its own rhythms—quite aside from their specialized activities. Living space and lived time are not worth a separate article in our encyclopedias. Our human space and time are a shambles, and that's perhaps why they are carefully avoided.